#6469693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1647 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,26,29,32)(21,25,22,31)(24,27,30,28);(9,10)(11,15)(12,13)(14,16)(17,19)(18,24)(20,30)(21,28)(22,27)(23,29)(25,32)(26,31);(2,6)(7,8)(10,13)(14,15)(17,20,23,18)(19,24,29,30)(21,25,22,31)(26,27,32,28);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,25,20,31)(19,27,29,28)(24,26,30,32);(1,4,5,3)(2,8,6,7)(17,21,23,22)(18,25,20,31)(19,28,29,27)(24,32,30,26)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376889);